Firstly, don’t assume yet that travel to Canada is going to be possible this May. Canada is likely to lag a little behind the US and the UK on Covid vaccine distribution due to a lack of domestic production capacity. We might be a little slower to re-open our borders to visitors.
Secondly, I am not sure if non-residents can insure or register their vehicles in Canada. There must be loopholes because I have read stories of European travellers buying and selling vehicles. If you do manage to purchase, register and insure a van, my suggestion would be to select a province with low insurance rates. Since you are landing in Montreal, getting a vehicle there would be ideal. Insurance rates in Quebec are low and safety standards are unfortunately lax so you could probably have little issue getting a number plate. That being said, you’ll still want a safe vehicle. Settlements especially north of Lake Superior in northern Ontario to the boundary with Manitoba are small and far apart. Breakdown services are going to be expensive and time-consuming. Public transit is poor to non-existent outside medium-sized cities.
Hiring a vehicle might be expensive. Figure on $40-80 per day, plus another $30 per day insurance and potential damage charges depending on what level of insurance you buy. One-way hires can be expensive and some companies will limit rentals to one month. Try a few international car-hire sites to check for yourself. Google “car rental horror stories Canada” to see how some companies exploit clients. Your ages might also make it difficult to hire a car. Car companies are hesitant to rent to under 25s.
Petrol is not expensive by UK standards. It is 91.5 cents per litre here in western Quebec, but the price is depressed due to Covid and reduced consumption. Expect to pay $1.50 per litre in Quebec and BC, $1.30 from Ontario to Alberta when commerce and travel and wide open. The distance from Montreal to Vancouver is about 5,000 km. At about 11 L/100 km fuel economy for a van, you are looking at consuming at least 550 litres of fuel just for travelling. That doesn’t include time idling in traffic, using the vehicle for heating or cooling and taking side trips for provisions or sight-seeing.
Also consider that depending on where you are, it can still be cold in May. Victoria Day weekend, which contains the third Monday in May is when Canadians start their camping season. We celebrate the birthday of a foreign queen on a day that is not her birthday by going into the bush and pretending that it’s not still cold. Temperatures often go below zero overnight even in the area where I spend most of my time which is the triangle formed by Montreal, Toronto and Ottawa. Personally, I don’t like when overnights go below 10, and I refuse to camp in temps that go below 5. You could get a propane heater, but they do have safety and humidity issues. Electric heaters would tether you to locations where you can plug in.
I am not trying to dissuade you from coming here. I can’t think of a better place to live or visit than this country. I would suggest if possible that you do the trip from June to August. Those extra few weeks make a large difference in weather. Outdoor swimming is possible by the second week of June. Also, if you can, keep a keen eye on changes to pandemic preparations and recovery. You may avoid a cancelled flight or refusal to enter at customs.